Also, check out table 8-1 on page 122 and "Movement in Combat" on page 130, column one at the bottom.
Basically running is your speed x4, or x3 in heavy armor or carrying a heavy load. You lose your dex bonus to AC and the square from where you started running is considered threatened even if you do nothing but run.
It would be the Run action (PHB, page 127). It is a full round action that allows you to move x4 your current Speed (or x3 if your are in heavy armor).
When you Run you can only go in a straight line, you lose your Dex bonus to AC, and the act of Running provokes an AoO. (So if you Run into someone's threatened area you immediately provoke an AoO, as indicated by the table on page 122 of the PHB.)